Simplify and expand the following expression: $ \dfrac{2k - 6}{k + 4}+\dfrac{k - 6}{3k - 6} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(k + 4)(3k - 6)$ Multiply the first term by $\dfrac{3k - 6}{3k - 6}$ $ \begin{align*} \dfrac{2k - 6}{k + 4} \times \dfrac{3k - 6}{3k - 6} & = \dfrac{(2k - 6)(3k - 6)}{(k + 4)(3k - 6)} \\ & = \dfrac{6k^2 - 30k + 36}{(k + 4)(3k - 6)}\end{align*} $ Multiply the second term by $\dfrac{k + 4}{k + 4}$ $ \begin{align*} \dfrac{k - 6}{3k - 6} \times \dfrac{k + 4}{k + 4} & = \dfrac{(k - 6)(k + 4)}{(3k - 6)(k + 4)} \\ & = \dfrac{k^2 - 2k - 24}{(3k - 6)(k + 4)}\end{align*} $ Now we have: $ = \dfrac{6k^2 - 30k + 36}{(k + 4)(3k - 6)} + \dfrac{k^2 - 2k - 24}{(3k - 6)(k + 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{6k^2 - 30k + 36 + k^2 - 2k - 24}{(k + 4)(3k - 6)} $ $ = \dfrac{7k^2 - 32k + 12}{(k + 4)(3k - 6)}$ Expand the denominator: $ = \dfrac{7k^2 - 32k + 12}{3k^2 + 6k - 24}$